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2 added 371 characters in body

I still find the statement of the problem very confusing. For $i=1$, you want your set $X$ to be a non-averaging set, that is, a set containing no three distinct elements $a,b,c$ such that $a+b=2c$. You want more than that, but that's a start, and there's enough literature on non-averaging sets to give you some kind of lower bound on $k$.

Tsuyoshi Ito posted an answer while I was typing mine, you'll see we're thinking along similar lines.

EDIT: There are several sections of Guy's Unsolved Problems In Number Theory that discuss problems not exactly what you want but not a million miles removed, either, and some of the references given there may be useful. Problem C8 is sets with distinct sums of subsets, C11 is three-subsets with distinct sums, C14 is maximal sum-free sets, C16 is nonaveraging sets.

I still find the statement of the problem very confusing. For $i=1$, you want your set $X$ to be a non-averaging set, that is, a set containing no three distinct elements $a,b,c$ such that $a+b=2c$. You want more than that, but that's a start, and there's enough literature on non-averaging sets to give you some kind of lower bound on $k$.